On the Extension of Surjective Isometries whose Domain is the Unit Sphere of a Space of Compact Operators

We prove that every surjective isometry from the unit sphere of the space K(H), of all compact operators on an arbitrary complex Hilbert space H, onto the unit sphere of an arbitrary real Banach space Y can be extended to a surjective real linear isometry from K(H) onto Y. This is probably the first example of an infinite dimensional non-commutative C*-algebra containing no unitaries and satisfying the Mazur- Ulam property. We also prove that all compact C*-algebras and all weakly compact JB*-triples satisfy the Mazur-Ulam property.